Moist PV
I was recently made aware of a paper by Wayne Schubert on Potential Vorticity (PV) in a moist, precipitating atmosphere:
Unfortunately this version is in a German journal so it's a little difficult to get a hold of. But a very similar version can be found in JAS from 2001:
This paper begins by deriving a generic form of a PV budget. They then chose the dry entropy or potential temperature as their thermodynamic variable which annihilates the solenoidal term (since these are a function of both density and pressure). This leaves the dry PV budget:
Here, a signifies that it's dry and P is PV while the other variables have their typical meanings.
One of my favourite paragraphs in this paper discusses the benefits of PV:
"It is sometimes argued that it is material conservation that makes PV so important. However, material conservation of PV is only valid away from the boundary layer and on short enough time scales that diabatic effects can be neglected. What really makes PV such a dynamically important quantity is not approximate material conservation, but rather the fact that it carries all the information about the balanced part of the wind and mass fields. In many geophysical flows the balanced part of the flow is by far the dominant part."
I agree strongly with this statement. Too often do I hear that the material conservation of PV is why it's important. Don't get me wrong, it's a useful property, but one other issue that Schubert does not point out is the difficulty of dealing with isentropic surfaces (as I've mentioned in previous blog posts). The usefulness of PV really comes from it's direct relation to balanced dynamics, as Schubert rightly points out. For anyone that doesn't use that phrase often, if we have balanced dynamics it fundamentally means that we can predict one field from changes in another. The most obvious example is geostrophic balance where changing the mass field will produce a geostrophic adjustment that over some time changes the wind field to be in balance. Unbalanced dynamics essentially mean that our best method of forecasting is stochastic methods. Thus, without balanced dynamics, deterministic models are essentially useless. Since PV relates the mass field to the wind field, it tells us the balanced part of the flow, and arguably that is it's most important quality. There is no other variable that I know of that can tell us so much about the important part of the flow.
Anyway back to the main point of the paper, the end result is actually fairly simple. In a moist precipitating atmosphere the PV has the same form but the virtual potential temperature takes the place of dry potential temperature. Further the diabatic and frictional terms in the PV budget use the virtual potential temperature, and there is a new term in the PV budget to account for precipitation and its differing fall speed to dry air and water-vapour:
Here subscript 'rho' indicates that you are taking the full density into account (i.e. including the density of moisture and precipitation), 'theta rho' is the virtual potential temperature, and U is the relative fall speed of precipitation.
The extra term can be interpreted as the relative flux divergence of precipitation (scaled by PV and the full density). Thus in the moist atmosphere there is also a source of PV from precipitation falling and varying the mass field. I haven't examined this yet using any data but I expect the precipitation mass flux to be strongly downward in the low-mid-levels of a deep convective cloud, maximizing at the lowest levels before precipitation falls out the bottom of the cloud and begins to evaporate. Thus the flux divergence of precipitation would be most negative (convergence) in the low-levels and most positive (divergence) in the upper-levels. This will result in a low-level sink of PV and an upper-level source of PV in association with a deep convective cloud.
Qualitatively this might offset the effects of diabatic heating since heating is strongly related to precipitation production and this typically produces the opposite dipole of PV sources and sinks (e.g.e source in the low-levels, sink in the upper-levels). The exact degree to which these terms offset each other is possibly going to be a small residual that's fraught with error, much like with the offset between the vertical advection and diabatic heating term.